The non-uniqueness of Lagrange functions is a standard topic in classical mechanics. However, it is still troublesome in undergraduate and graduate teaching. In this paper, we show that this non-uniqueness is equivalent to gauge freedom, and thus the gauge, canonical, and Galilean transformations are equivalent. In this sense, we also discuss how to solve the problem of selecting one out of many gauge-equivalent Hamiltonians to represent energy.

中文翻译：

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规范，规范和伽利略变换以及拉格朗日函数的非唯一性

拉格朗日函数的非唯一性是经典力学中的一个标准主题。但是，在本科生和研究生教学中仍然很麻烦。在本文中，我们证明了这种非唯一性等同于规范自由度，因此规范，规范和伽利略变换是等效的。从这个意义上讲，我们还将讨论如何解决从众多等价的哈密顿量中选择一个来代表能量的问题。